Let $A\colon Q_{\mathbb{R}}\to N_{\mathbb{R}}$ be an affine map defined as $A=H+u_{0}$ for an injective linear map $H$ and a point $u_{0}\in N_{\mathbb{R}}$. Let $f\colon C\to\mathbb{R}$ be a concave function of Legendre type defined on an open convex set $C\subset N_{\mathbb{R}}$ such that $C\cap\operatorname{im}(A)\not=\emptyset$. Then $A^{*}f$ is a concave function of Legendre type on $A^{-1}(C)$,

$$\operatorname{stab}(A^{\ast}f)^{\circ}=\operatorname{im}(\nabla(A^{*}f))=H^{\vee}(\operatorname{im}(\nabla f))=H^{\vee}(\operatorname{stab}(f)^{\circ}),$$

and, for all $v\in A^{-1}C$,

$$(A^{\ast}f)^{\vee}(\nabla(A^{\ast}f)(v))=f^{\vee}(\nabla f(Av))-\langle\nabla f(Av),u_{0}\rangle.$$

Moreover, there is a section $\imath_{A,f}$ of $H^{\vee}|_{\operatorname{stab}(f)^{\circ}}$ such that the diagram

(3.56)

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commutes.